In this section, we derive the two-point correlation function for systems of polydisperse, totally penetrable spheres in the first three dimensions following the approach in Refs. [1]}, [2]}, [3]}, [4]}. We take sphere radii \(R\)  to follow the normalized probability density \(f(R)\) . The average of any \(R\) -dependent function is thus computed as
\(\langle w(R) \rangle = \int _0^{\infty } w(R)f(R)dR.\) 
